Moral Hazard and the US Stock Market: Has Mr Greenspan Created a Bubble?

Transcription

1 Moral Hazard and the US Stock Market: Has Mr Greensan Created a Bubble? Marcus Miller eartment o Economics and CSGR University o Warwick and CEPR Paul Weller eartment o Finance University o Iowa Lei Zhang eartment o Economics University o Warwick ecember 1999 Acknowledgement: We would like to thank articiants to seminars at the Bank o Finland and the Warwick Financial Otion Research Centre or their comments and suggestions, articularly those rom aniel Cohen and Stewart Hodges. The aer was comleted when Marcus Miller was Visiting Scholar at the IMF Research eartment and he is grateul or their hositality. The views exressed are those o the authors, and do not reresent those o the IMF.

2 Abstract The current risk remium in the US stock market is ar below its historic level and the market continues to rise.. With the long run real interest rate not much, i at all, higher than the growth rate, dividend valuations become very sensitive to variations in the remium: and we show how the latter can be reduced by one-sided intervention olicy by the Fed which lulls investors into a alse sense o security. We assume investors value their ortolios as i they held a ut otion, with an exercise rice 25% below the revious market eak. Since the Fed cannot determine the real value o stocks, the resulting asset rices are not rational, so our account involves a degree o myoia and over-otimism on the art o the average investor. We rovide calibrations to demonstrate that the "sliding ut" can reconcile booming stock rices with unchanged risk remia. Although there are some good reasons why risk remia may have allen below the long run average o 7%, by showing the owerul eect that changing ercetions o down-side risk can exert on asset rices, we have strengthened the case or treating current asset valuations with susicion. 2

3 Beaucou d investisseurs ont conclu que les cours boursiers ne euvent que monter. Blanchard (1999) 1. Introduction Since the stock market break o 1987, shares in the US market have areciated at a recordbreaking ace. The S & P 500 index, or examle, has increased rom about 220 in October 1987 to a little over 1,400 now, an increase o over 500% or an average annual growth rate o about 17%. This asset rice boom imlies that relative to the ast, estimated growth rates have risen, the risk remium has allen, or there is a bubble (or some combination o the three). With the high technology sector as market leader, there has been much discussion o aster than exected growth based on the new communications technology. This is not the ocus o this aer. What we do here is to investigate the all in the risk remium in the US stock market, taking as given the (airly conservative) growth orecasts o around 3% made by the economists we cite. In articular, we suggest that the aarent all in the remium could be a sohisticated asset bubble. The idea we develo is what Blanchard alludes to in the quotation above rom his discussion o market develoments beore the French Council o Economic Advisers: that many investors are convinced that the market can only go u! Why should this be so? And how could it aect the market? The reason, we suggest, is a orm o moral hazard. Investors in the US have become convinced that the Federal Reserve will take decisive action to revent the market alling but not to sto it rising: and 3

4 they believe that these actions will work. So the Fed is aarently insuring them. The eect is like a ut: but the reality is a bubble, because the ut will not exist when it comes to be exercised. Key ieces o evidence are the actions taken by Mr Greensan in halting the market break o 1987 and in checking the market all in the liquidity crunch o 1998, in both cases by cutting interest rates and uming in liquidity. The monetary authority cannot control the real interest rate in the long run, but it can over the short run when rices and inlation exectations are sticky. So it can aect share rices, at least or a while. By correcting one crash and averting another Mr Greensan has done enough to ersuade investors that he will stabilise the market long enough or them to get out, keeing the gains they have made to date. I they all sell, the market will crash: so the logic is allacious. But the gains are temting and even Mr Greensan is beginning to talk o a new aradigm or the US economy. Historically the risk remium has been estimated to lie between 7% and 8%. But consider the two estimates o the risk remium in the US in 1999 shown in Table 1. These are arrived at by simly subtracting the risk ree real interest rate rom the total yield (dividend yield lus growth). In his comments to the French CEA (between Aril and June), Blanchard ut the remium at 2% in the second quarter o A ew months later, in Setember, Wadhwani, a member o Britain s Monetary Policy Committee who had written a widely cited aer on the US bubble in 1998, reckoned the remium was down to only 1%. As can be seen rom the table the reason or the one oint lower ividend/rice Ratio (%) ividend growth Rate (%) Real interest Rate Risk remium (%) Author 4

5 Blanchard (Q2, Wadhwani (Q3, 1999) Table 1 Two recent estimates o equity risk remium in the US stock market In his comments, Blanchard acknowledged that there are good reasons or the risk remium being lower than in the ast, but thought 2% is too low and exressed the view that there is a bubble, arising artly rom overestimating growth, artly rom ure extraolative ricing exectations. In his NIER aer, Wadhwani (1998, see also Financial Times, 18-Se-1999) also dismissed the idea that the risk remium had virtually disaeared. He noted that i you ut the remium at 3%, and kee the growth igures this gives estimated over-valuation o about 100% (i.e., a all o 50%). Beore develoing our exlanation, we summarise the lively debate on the equity risk remium in the next section o the aer. Then we develo our account o the bubble, namely that the asymmetric behaviour o the monetary authorities has established a loor to market rices, but no ceiling: and that this loor ratchets u whenever the market reaches a new eak. More recisely, stocks are riced as i market articiants were in ossession o an undated ut with an exercise rice 25% below the last eak. The idea o monetary intervention having rice eects like the issue o derivatives is amiliar rom the work o Paul Krugman (1991) on target zones or exchange rates: this is a one sided target zone or the stock market. While Krugman s target zone or the nominal rate deended on the authorities having enough reserves, a erceived loor on the real rice o stocks requires gullibility and myoia o the art o the 5

6 average investor. Linking the exercise rice to ast eaks is a eature o stock market trading rules exlored by Grossman and Zhou (1993); in circumstances where there is a ositive trend in undamentals it adds greatly to the value o a ut. We show theoretically how the erceived ut raises rices and reduces the imlied risk remium: and we rove there exists a unique cone that market rices are restricted to, susended well above their undamental value. Then we calibrate the model using arameters rom the table above and the risk remium set at its ost-war average. We indthat a sliding ut allowing or a 25% all could bring estimated risk remia down rom 7% to 2%, i.e., that it could account or the current low values reorted above, even though the underlying arameters are at their historical values. The imlication is that the market has a long way to all when undamentals drag rices ar enough down or the average investor to try exercising. With overvaluation o over 100%, the all is bigger than 50%. (These results deend on the ut being comletely credible, and will be less dramatic as this assumtion is weakened.) Beore concluding, we suggest how to reconcile our aroach with the rice o uts on the market which notoriously charges a lot or uts that are ar out o the money. Those buying and selling uts do not share the oular view, but are not big enough to change it. 2. The Equity Premium The essence o the equity remium uzzle identiied by Mehra and Prescott (1985) is that, in a reresentative agent asset ricing model, it is necessary to assume an imlausibly high degree o risk aversion in order to reroduce the historical level o the remium. The reason or this is that the risk remium in such a model is determined by the covariance between consumtion growth and the return on 6

7 the stock market multilied by the coeicient o relative risk aversion. Since consumtion growth has an annual standard deviation o about one er cent, the covariance is small, and this translates into a large value or the measure o risk aversion i one is to match the value o the equity remium in ost-war US data. Cambell (1998) reorts a value o 7.85% or the eriod There have been numerous attemts to exlain the equity remium. We grou them into ive broad categories, which are not necessarily mutually exclusive. The irst category contains models that aim to make the assumtion o high risk aversion more lausible. Cambell and Cochrane (1999) construct a model with time-varying risk aversion driven by habit ersistence. When consumtion alls close to the level o the habit, or examle in recession, individuals become highly risk averse. But in eriods o exansion risk aversion alls. One o the attractive eatures o this model is its ability to match a number o other eatures o the data or which the standard model ails. Hansen, Sargent and Tallarini (1997) describe an economy in which the reresentative agent is assumed to be ignorant o the true model generating stock rices. They seciy a orm or the utility unction in which high risk aversion can alternatively be interreted as a reerence or robustness to small seciication errors. The second category argues that the objective uncertainty o stock market returns is greater than is revealed in the samle data. Rietz (1988) shows that introducing a small robability o a large negative shock to consumtion growth is suicient to exlain the remium. The robability can be made suiciently small that there would be little chance o observing such a shock even in data sanning a century. Brown, Goetzmann and Ross (1995) observe that in the ast history o some major markets other than the US Russia, China, Germany and Jaan there have been one or more major interrutions that lead to their being excluded rom long term studies o stock returns. This orm o samle selection bias may lead to a 7

8 substantial underestimation o the true risk o the stock market, a risk that is correctly erceived by investors. The third category introduces various deartures rom strict or narrow rationality. In Kurz and Motolese (1999) the resence o endogenously determined dierences o oinion consistent with a weaker notion o rationality can account or the remium. Cecchetti, Lam and Mark (1998) modiy an otherwise standard Lucas asset ricing model by assuming that individuals miserceive the ersistence o exansions and contractions. They show that i individuals believe that both exansions and contractions are less ersistent than is revealed by the data, and i these belies exhibit random variation over time, then one can reroduce the level and volatility o the equity remium. The ourth category introduces heterogeneity among investors. Constantinides and uie (1996) construct a ramework in which there is cross-sectional variation in consumtion. They show that i the cross-sectional variance o log consumtion growth is negatively correlated with the level o aggregate consumtion, so that individual consumtion risk increases in recessions, this can hel exlain the excess returns to stocks without invoking high levels o risk aversion. The ith category introduces rictions. Heaton and Lucas (1996) and Krusell and Smith (1997) ind that it is necessary to ostulate large costs o trading equities or borrowing constraints to exlain the equity remium. Marshall and Parekh show that very small ixed costs o adjusting non-durable consumtion are caable o exlaining some, but not all o the equity remium. Their inding is exlained by the act that at the otimum the utility gains rom adjusting consumtion in resonse to changes in asset returns are small. Some have argued that the standard model can be used to rationalise the current market valuation. In the light o the research summarised above, this seems unconvincing. Although no single 8

9 model rovides a ully satisactory exlanation or the equity remium on its own, many suly ersuasive arguments as to why one should not exect the standard model to match the data. In addition, or the standard model to rationalise current stock valuations it is not suicient simly to observe that the level o the market is now consistent with a lausible level o risk aversion. The model has to exlain why there has been a reciitous dro in the level o risk aversion over the sace o a ew years, and why this henomenon has been largely conined to the US. This it consicuously ails to do. The habit ersistence model o Cambell and Cochrane (1999) can roduce eriods during which the rice-dividend ratio is high, but a necessary condition or this is high consumtion growth in the recent ast. This does not characterise the US exerience over the last decade. Models such as that o Rietz (1988) could in rincile tell a story in which agents suddenly come to believe that the risk o a serious market collase no longer exists. However, in a world o rational learning, changes in belies are generally rather gradual. The sudden change is diicult to rationalise other than in an irrational world. 3. The Model We consider the roblem acing a reresentative investor who can trade an asset which ays dividends at the rate ( t) dt. ividends are assumed to evolve according to: where d = µ dt σ dz, (1) µ is the trend, z is a standard Brownian motion and σ the standard deviation. The rice o the asset, V ( ), then ollows a diusion rocess: dv = µ (.) dt σ ()dz.. (2) V 9

10 The notation (.) indicates that both drit and volatility can be unctions o the state variable (t). The instantaneous total return on the asset is given by: dv dt, (3) V V where the irst term indicates caital gain and the second the dividend rate. The utility o the investor is: 0 E0 u )] δt [ c( t e dt. (4). where E 0 is the exectations oerator conditional on time 0, u ( ) is the instantaneous utility unction, c (t) the consumtion and δ the rate o time reerences. I we deine a stochastic discount actor as M t ( t) e u [ c( t)] = δ, (5) then the investor s Euler equation in continuous time takes the orm o This can be rewritten as [ d( MV )] 0 = Mdt E. (6) t dm dv dm dv 0 = dt Et. (7) V M V M V I we aly the above equation to a risk ree asset, we obtain an exression or the risk ree rate o interest: r dt dm = Et. (8) M Substituting (8) into (7) yields: 10

11 dv dm dv Et dt = r dt Et. (9) V V M V The last term on the right hand side reresents the risk remium. We assume that the utility unction has the orm: and that consumtion evolves as: ( t) 1γ c u [ c( t)] = (10) 1 γ dc c = µ dt dz. (11) C σ C Treating V as a unction o dividends and alying Ito s Lemma gives the ollowing exression or the risk remium. dm Et M dv V = C V γρσ σ (12) V where ρ is the correlation between consumtion growth and dividend growth, and the rime indicates derivatives. Alying Ito s Lemma to (9) and substituting in the exression in (12) gives 1 2 ( ) ( µ γρσ σ ) V ( ) r V( ) = 0 2 σ V C. (13) 2 This equation has the linear solution V ( ) = r µ γρσ σ. (14) C 11

12 This is just a continuous time version o the amiliar Gordon growth model ormula. In act, one can derive equation (14) by treating the asset rice, V (), as the exected resent value o all current and uture dividends discounted by the risk adjusted rate o r = r γρσ cσ, i.e., where = rt V( ) E0 ( t) e dt =, (15) r µ π π = γρσ σ c 0 is the true risk remium, as distinct rom the risk remium in (12) which is inluenced by alse investor belies about the eects o intervention by the Federal Reserve. However, we are interested in a class o non-linear solutions to (13) that arise as a consequence o such belies about intervention. Seciically, we interret V ( ) as the value o the market ortolio and suose that investors believe that the Federal Reserve will adjust monetary olicy to suort the market whenever it has allen to a level o? times its revious maximum level. In what ollows, we show how a belie in the eectiveness o such a olicy can create stock rice bubbles. 4. Asymmetric Monetary Policy, Moral Hazard And Stock Price Bubbles Assume that monetary olicy is conducted in such a way that real rates are unchanged when stocks rise, but that nominal (and short run real) interest rates are cut whenever stock rices all to a raction η o the revious market eak. I investors exect that this will stabilise rices or long enough to exit the market, it is as i they have ree ut otions insuring them against downside risks. With this asymmetry o monetary olicy built into exectations, stock rices will be substantially inlated. In this sector we characterise these bubble solutions. Let the maximum value o the market u to time t be 12

13 S t = { Max{ V ( τ )}, τ t}. (16) I the stock rice lies in the range η S t, S ), then its value is determined by equation (13), with general solution as ollows ( t ξ = ( A A, (17) r µ V ) ξ where µ = µ π, is the dividend level at S t, A and A are two constants to be determined, and ξ and ξ are the ositive and negative roots o the quadratic equation 1 2 σ ( 1) ξ ξ µ ξ r = 0. (18) 2 It can be shown that ξ > 1 and ξ < 0. elating dividends by in the second and third terms on the right hand side o (17) simliies the subsequent algebra but does not aect the solution. I stock rices reach η St matching and smooth asting conditions must hold: where, stabilisation is assumed to occur. This imlies that the ollowing value V ( b ) = ηs, (19) b V ( ) = 0, (20) b is the lower dividend level corresonding to the level o stock rices at which investors believe the market will be stabilised. At a market eak, no change o olicy is exected. The deinition o imlies that V( ) = S. (21) 13

14 I dividends move above, then a new market eak is attained and the solution in (17) is revised uwards conditional on this new eak. I dividends move below, then stock rices will be determined by the solution to (17). Let all eaks be given by the enveloe S = ( ). (22) The boundary condition at the eak imlies that V ) = ( ). (23) ( Then we can state the ollowing result. Proosition For 0 η< η max < 1, there exists a unique solution α 0 such that α ( ) = 1, (24) r µ The solution satisies the ollowing conditions: and α η > 0, (25) lim α( η) = 0 η 0 lim α( η). (26) η ηmax Proo: See Aendix A. One way o understanding the linearity o the enveloe established in the roosition is to introduce the transormation y / =. Then we see rom (13) that v y) V ( / )/ ( = satisies the same 14

15 equation. It is natural to conjecture that the boundary condition or v( y) will be homogeneous o degree zero with resect to boundary values o y. This suggests that the enveloe must be a linear unction o. Since the enveloe or all eaks is given by (24), the enveloe or all stabilisation oints will also be a linear unction. Seciically, as ηs = η( 1 α) η ( 1 α) = * r µ x r µ b, (27) so the enveloe or all stabilisation oints is η ( 1 α) e( b ) = * x r µ b, (28) where x * is the solution to (A9) in Aendix A and η / * x >1 as shown in Aendix B. These two enveloes orm a cone within which all solutions conditional on a given value o will lie. We use Figure 1 to illustrate one o these solutions. The undamental solution as in (15) is shown as the lowest straight line rom the origin. The two enveloes, which orm the cone, are given by ( ) and e( b ) or all eaks and all stabilisation oints resectively. The solution or the stock rice conditional on in (17) is reresented by the convex curve V, which smooth astes to a horizontal line at the bottom and smooth astes to the enveloe ( ) at. All other short run solutions will resemble V. As the conditional solutions are lat at the stabilisation oints and steadily rise towards eaks, the stock rice volatility is lower when the stock rice is lower and increases as the stock rice rises. (Note that the instantaneous variance o the stock rice deends on the sloe o the conditional solution.) 15

16 Stock valuation e( b ) ( ) S ηs V Fundamental Value b ividends Figure 1. Asymmetric monetary olicy, moral hazard and stock rice bubbles. We see rom (12) that our model redicts a risk remium which deends on and. We roceed to calculate the uer and lower bounds occurring at b and. Using the ormula in (15), one can show that the imlied risk remium at eaks is given by π π = α( µ r 1 α ). (29) This gives the uer bound or the imlied market risk remium. The lower bound is derived using the enveloe or stabilisation oints which gives π b * * π [ 1 (1 α) η / x ]( r µ ) x =. (30) 1 α η * The over-valuation at market eaks is given by α, while at stabilisation oints it is ( 1 α) η / x 1. In the next section, we use numerical method to look at these measures. 16

17 5. Numerical Results The arameter values we use or the base line case below are as ollows: the real interest rate r = 0. 03, the true risk remium π = 0. 07, the dividend growth rate µ = 0. 03, the volatility o stock rices σ = Stabilisation is assumed to occur when stock rices are 25% below the revious eak, so η = To examine the sensitivity o the results to our choice o arameter values we vary the real interest rate rom 0.02 to 0.04, the risk remium rom 0.06 to 0.08, and the volatility o stock rices rom 0.15 to The numerical results below show the maximum ossible value or η, η max ; the imlied uer and lower bounds o risk remia, π and π b, and the over-valuation o stock rices at eaks. Risk remium Maximum Uer bound Lower bound Over-valuation Stabilisation η max π π b α 6% % 1.26% % % 1.61% % % 1.89% 1.63 Table 2: The Eect o Changing Stock Price Risk Premium (r 0.03, σ = 0.2, µ = 0.03). 17

18 Price Maximum Uer bound Lower bound Over-valuation Volatilityσ Stabilisation η max π π b α % 2.33% % 1.61% % 0.44% Table 3. Eects o Changing Stock Price Volatility (r =0.03, π =0.07, µ =0.03). Interest rate Maximum Uer bound Lower bound Over-valuation r Stabilisation η max π π b α 2% % 2.14% % % 1.61% % % 1.02% 1.40 Table 4. Eects o Changing Real Interest Rate ( π=0.07, σ=0.2, µ =0.03). Tables 2-4 illustrate how changing the risk remium, the volatility o stock rices and the real interest rate aect imlied risk remia and stock rice over-valuations. In almost all cases, a stabilisation rule imlying that intervention occurs when stock rices have allen by 25% rom their revious eak can reduce the historical risk remium by more than hal. 18

19 Table 2 shows how changing the true risk remium aects the imlied risk remium. The third and the ourth columns show that the imlied risk remium goes u with the true risk remium, while the degree o stock rice over-valuation goes down. Since higher π means lower α, rom (29) both o these two eects increase the imlied risk remium. Table 3 illustrates the imact on the results when stock rice volatility is increased. Higher volatility means that, or a given stabilisation rule, the insurance value is higher. This ushes u α and so reduces the imlied risk remium. Table 4 shows the eect o changing the real interest rate. Increasing the real interest rate decreases the value or α, which translates into a higher imlied risk remium. But a higher interest rate has a direct negative eect on the imlied risk remium. The simulation results in Table 4 show that this direct eect dominates. Note inally that the numerical results are consistent with the risk remia estimated in Table 1. The highlighted row in Table 2 shows a case similar to that o Blanchard (1999). With the same real interest rate and dividend growth rate as in Blanchard, the average o the two imlied risk remia rom our simulation is 2%, exactly the same as Blanchard s estimate. 6. Why Are Out-O-Money Puts So Exensive? I we look to the otions market or suort or our thesis, we ace an obvious roblem. It is well known that imlied volatility or ut otions is higher or low values o undamentals. At irst sight, this aears to reute the model we are roosing. I the Federal Reserve is believed to be insuring asset rices or ree, why should rivate insurance be so exensive? With ully rational investors otion rices do indeed relect true risk-neutral robabilities. But we are describing a world inhabited by a majority o 19

20 investors who have one seciic irrational belie, namely that the Fed has the ower to stabilise the market once it has allen by more than a certain amount. Sohisticated investors do not believe this. Thus they believe that the market is overvalued and have an incentive to buy crash insurance. This insurance is sold by other sohisticated investors who rice the uts accordingly. O course, the grou o sohisticated investors has to be in a suiciently small minority that their trades in stocks do not have a signiicant imact on the market. I, as one resumes should be the case, some ortolio managers are included in this grou, we can aeal to the arguments o Shleier and Vishny (1997) or an exlanation as to why their belies are not relected in market rices. Fund managers are discouraged rom taking large bets against the market because they know that in the short run i rices move against them i.e. urther rom undamentals, they will suer cash withdrawals and be less able to exloit what is now a more avourable investment situation. This still leaves oen the question o why the irrational investors do not sell ut otions which they consider to be overriced. We suggest that there are several reasons or this. Many such investors delegate the task o ortolio management to mutual und and ension und managers, who are tightly restricted in their trading o derivatives. These managers may also not believe in the ower o the Fed. In addition, otions are more comlex inancial instruments than stocks, whose returns are much less transarent. It should also be remembered that we are describing a situation that cannot last orever, and that is likely to be rather short-lived. By the time the average investor has realised that his belie in the ower o the Fed imlies unexloited roit oortunities in the otions market, those oortunities will robably no longer exist. 20

21 7. Conclusion Both economists cited in the Introduction assume that the long run real interest rate (3 to 4 ercent) is not much, i at all, higher than the growth rate (say 3 ercent). This means that dividend valuation is extraordinarily sensitive to the estimated risk remium: and we have shown how the risk remium can be reduced by one-sided intervention olicy by the Fed which lulls eole into a alse sense o security. Since the Fed cannot determine the real value o stocks, the resulting asset rices are not rational, and our account admittedly involves a degree o myoia and over-otimism on the art o the average investor. Myoia is needed so that temorary changes can be treated as long-lasting, and (with short run cuts in real rates treated as ersistent) the ower o the monetary authorities over asset rices is exaggerated. But even i the Fed cannot hold rates down or ever, could it not stabilise rices long enough or you to get out irst? Over-otimism is needed or the average investor to believe that. The calibrations demonstrate that our account could reconcile booming stock rices with very high underlying risk remia. But we do not, in act, want to claim that it is only mistaken belies about monetary olicy that exlain current high valuations. There are, as Blanchard remarked, some good reasons why risk remia may have allen below the long run average o 7% -- better stabilisation o the economy ("ine-tuning") and more eicient distribution o risk ("inancial engineering") being two. And, i investors are myoic, they can extraolate short run surges o growth into the long run (as seems to be true o internet stocks). By showing the owerul eect that changing ercetions o down-side risk can exert on asset rices, we have strengthened the case or treating current asset valuations with susicion. In their account o intrinsic bubbles, Froot and Obsteld (1991) aealed to the idea that the market might select the 21

22 wrong stochastic solution. 1 But they do not say why. Our sliding otion is, like theirs, the "wrong" solution. But it is the kind o bubble you can almost believe in. Reerences Blanchard, O.(1999). Commentaire, Architecture Financiere Internationale, 130. Les Raorts du Conseil d Analyse Economique, no.18, CAE. Brown, S. W. Goetzmann and S. Ross, (1995), "Survival", Journal o Finance 50, Cambell, J.Y. (1998), "Asset Prices, Consumtion and the Business Cycle", NBER Working Paer Cambell, J.Y. and J.H. Cochrane (1999), "Force o Habit: A Consumtion-Based Exlanation o Aggregate Stock Market Behavior", Journal o Political Economy 107, Cecchetti, S.G., P.-S. Lam and N.C.Mark, (1998), "Asset Pricing with istorted Belies: Are Equity Returns Too Good to be True?", NBER Working Paer Constantinides, G. and. uie (1996), "Asset Pricing with Heterogeneous Consumers", Journal o Political Economy 104, The same idea was also exlored in Miller and Weller (1990). 22

23 riill, J. and M. Sola (1998), Intrinsic Bubbles and Regime-switching, Journal o Monetary Economics, 42, Financial Times, (1999), Equity risk remium LEX COLUMN, 18-Se Froot, F. A. and M. Obsteld (1991), "Intrinsic bubbles: The case o stock rices", American Economic Review, 81, Grossman S. J. and Zhou Z. (1993), Otimal investment or controlling drawdowns, Mathematical inance, 3(3), Hansen, L.P., T.J. Sargent and T.. Tallarini (1997), "Robust Permanent Income and Pricing", unublished aer, University o Chicago and Carnegie Mellon University. Heaton, J. and. Lucas (1996), "Evaluating the Eects o Incomlete Markets on Risk Sharing and Asset Pricing", Journal o Political Economy 104, Krugman, P.(1991), Target Zones and Exchange Rate ynamics, Quarterly Journal o Economics, 56, 3,

24 Krusell P. and A.A. Smith, Jr. (1997), "Income and Wealth Heterogeneity, Portolio Choice, and Equilibrium Asset Returns", unublished aer, University o Rochester and Carnegie Mellon University. Kurz, M. and M. Motolese (1999), "Endogenous Uncertainty and Market Volatility", unublished aer, Stanord University. Marshall,.A. and N.G. Parekh (1999), "Can Costs o Consumtion Adjustment Exlain Asset Pricing Puzzles?", Journal o Finance 54, Mehra, R. and E. Prescott (1985), "The Equity Premium Puzzle", Journal o Monetary Economics 15, Miller, M. and P. Weller (1990), " Currency bubbles which aect undamentals: a qualitative treament ", Economic Journal, 1001(400), Rietz, T. (1988), "The Equity Risk Premium: A Solution?", Journal o Monetary Economics 21, Shleier, Andrei and Robert W. Vishny, The Limits o Arbitrage, Journal o Finance 52, 1997,

25 Wadhwani, Sushil (1998), The Valuation o US Stocks, NIER, November. Aendices A. Proo o Proosition Substituting boundary conditions (19) and (21) into (17) one can solve or A and A as ξ ξ x x x r A = ( )( η ) ( ) / ( µ ) ξ ξ x x (A1) and ξ ξ x x x r A = ( )( η ) ( ) / ( µ ) ξ ξ, (A2) x x where x = /. Substituting (A1) and (A2) into (20) yields b ξ ξ 1 ξ ξ 1 ξ ξ ξ x x x x x x x x ( ) ξ ( ) ( ) ( ) = ξ ξ 1 ξ ξ 1 ξ ( x η) x ξ ( x η ) x r µ = g( x) r µ (A3) Substituting (A1) and (A2) into (22) and rearranging yields ( ) = r ξ ξ ξ ξ 1 ξ ( x x) ξ ( x x) ξ ( x η) ξ ( x η) ( ) 1 ξ ξ ξ ξ µ x x x x (A4) It can be shown that ( ) ( ) ( ) = x x. (A5) 25

26 Substituting (A5) into (A4), one obtain a irst order linear dierential equation or ( ) / which has a solution as ollows ( ) = h( x). (A6) Combining (A6) and (A3), we have a ixed oint equation or x as h( x) = r g( x) µ. (A7) As long as there exists a solution x * * to (A7), g( x ) will be a constant. Let * g( x ) = 1 α, (A8) we have the linear enveloe given in Proosition 1. (We will show later there indeed exists a ixed oint to (A7).) Comaring the stock rice with exected stabilisation and without (as in equation (15)), the ormer is always greater than or equal to the latter, so α 0. Substituting (24) into (A3) and (A4), and eliminating α yields the ollowing ixed oint equation or x : ξ ξ ( ξ 1)( 1 ξ )( x x ) η = ξ ξ ξ ξ ( ξ 1) x ξ ( ξ 1) x 1 ξ 1 k( x ). (A9) Since 0 x 1, we only have to look at the roerty o unction k( x) or x [ 01, ]. It is straight orward to show that lim k( x) = 0 x 0 lim k( x) = 1 x 1 (A10) and 26

27 ξ 1 ξ 1 ξ 1 ξ 1 ( ξ 1)( 1 ξ )( ξ x ξ x )[ ξ ξ ( ξ 1) x ( ξ 1) x ] k ( x) = > 0 ξ ξ [ ξ ξ ξ ( ξ 1) x ξ ( ξ 1) x ] (A11) So or 0 η < 1 there exists a unique solution x * to (A9). In articular, i η = 0 then x * = 0. Substituting (24) into (A3) and using (A9) to relace η yields ξ ξ ξ ξ α = ξ ξ ξ ( ξ 1) x 1 ξ ( ξ 1) x 1 n( x). (A12) As n ( x) = ( ξ )( ξ )( ξ x ξ ξ ξ x ) x < 0 (A13) i n( xmax ) = 0 then n( x) 0, i 0 x x max; n( x) < 0, i x max < x 1. (A14) Since α 0, so 0 x x max, where x max 1/( ξ ξ ) ξ =. ξ 1 < 1 ξ 1 ξ (A15) From (A9), this imlies an uer bound or η such that a solution to α exists. Substituting (A15) into (A9) yields η max ξ 1 ξ = < 1 max ξ x. (A16) From (A11) and (A12), it can be shown that 27

28 α ξ ξ = η n ( x) n' ( x) dx >. (A17) dη 0 2 And lim lim η ηmax α = lim α = x α = lim α. (A18) 0 η 0 0 x xmax B. The Enveloe or Stabilisation Points To rove η/ * x > 1, rom (A9) we only need to show ξ 1 ξ 1 η ( ξ 1)( 1 ξ )( x x ) = > 1, (B1) ξ 1 ξ 1 x ξ ξ ξ ( ξ 1) x ξ ( ξ 1) x or equivalently to show m( x) ( ξ ) x ξ 1 ξ 1 1 ( ξ 1) x > ξ ξ As m( x) is strictly decreasing and then η/. (B2) m( xmax) ξ ξ, (B3) * x > 1. 28